Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

(propose a free ad)

Consider the period $T$ of the bound states of a nonlinear potential well $U(x)$. So $x(t)$ is just bouncing back and forth between $x_0$ and $x_1$.

$T(x_0)\propto\int_{x_0}^{x_1}\frac{\mathrm{d}x}{\sqrt{E-U(x)}}$ where $E=U(x_0)=U(x_1)$.

How do we prove $T$ is differentiable in energy i.e. $\mathrm{d}T/\mathrm{d}x_0$ exists at least for bound states. Leibniz integral rule doesn't really apply here. Do I have to go into the business of action-angle? Thanks.

Write $f(x_0,E,\delta)$ for the integral, taken between $x_0+\delta$ and $x_1-\delta$, where $\delta>0$ is tiny. Then you may differentiate under the integral with respect to E to get $\frac{df}{dE}(x_0,E,\delta)$. Then you must show that the limit of the result remains well-defined when $\delta\to 0$ to get $\frac{df}{dE}(x_0,E,0)$. Now you may use the chain rule to calculate the derivative of $T(x_0)=f(x_0,U(x_0),0)$.

So basically you treat $E$ as an independent variable in the first place, evaluate the integral, then plug in $E=U(x_0)$ and differentiate in $x_0$ via chain rule. Sounds brilliant. Thanks.

I was thinking of transforming the integration domain of any $T(x_0)$ to a fixed interval say $(0,\frac{1}{2})$ via change of variable. Then show that the derivative of the new integrand with respect to $x_0$ exists a.e and integrable.

user contributions licensed under cc by-sa 3.0 with attribution required